How do I get the glass material to stop weirdly showing the object inside the glass?
The problem you are having is that if you set no thickness to the glass then your object becomes a solid piece of glass:



If you give the glass some thickness (in this example I used a solidify modifier) then it will behave like an object that has glass panes, but is empty inside.



But the problem is not solved, because the default glass shader has problems dealing with direct light going through it.

Using the tricky glass shader detailed in this post: How to illuminate the darkness inside glass objects in Cycles? will give you more realistic results.